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Direct and Inverse Variation Problems

Learn how to solve direct and inverse variation problems in this chapter. Understand the concepts of direct variation and inverse variation, write variation equations, substitute given values, and find the required answers. Examples and step-by-step solutions included.

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Direct and Inverse Variation Problems

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  1. Chapter 7 Section 8

  2. Variation Solve direct variation problems. Solve inverse variation problems. 7.8 2

  3. Objective 1 Solve direct variation problems. Slide 7.8-3

  4. Two variables vary directly if one is a constant multiple of the other. Direct Variation y varies directly as x if there exists a constant k such that Solve direct variation problems. In these equations,y is said to be proportionaltox. The constant kin the equation for direct variation is a numerical value. This value is called the constant of variation. Some simple examples of variation include: Direct Variation: The harder one pushes on a car’s gas pedal, the faster the car goes. Inverse Variation: The harder one pushes on a car’s brake pedal, the slower the car goes. Slide 7.8-4

  5. Solving a Variation Problem Step 1:Write the variation equation. Step 2:Substitute the appropriate given values and solve for k. Step 3:Rewrite the variation equation with the value of k from Step 2. Step 4:Substitute the remaining values, solve for the unknown, and find the required answer. Solve direct variation problems. (cont’d) Slide 7.8-5

  6. If z varies directly as t, and z = 11 when t = 4, find z when t = 32. EXAMPLE 1 Using Direct Variation Solution: Slide 7.8-6

  7. Direct Variation as a Power y varies directly as the nth power of x if there exists a real number k such that Solve direct variation problems. (cont’d) The direct variation equation y = kx is a linear equation. Other kinds of variation involve other types of equations. Slide 7.8-7

  8. The circumference of a circle varies directly as the radius. A circle with a radius of 7 cm has a circumference of 43.96 cm. Find the circumference if the radius is 11 cm. EXAMPLE 2 Solving a Direct Variation Problem Solution: Thus, the circumference of the circle is 69.08 cm if the radius equals 11 cm. Slide 7.8-8

  9. Objective 2 Solve inverse variation problems. Slide 7.8-9

  10. Inverse Variation y varies inversely as x if there exists a real number k such that Also,y varies inversely as the nth power of x if there exists a real number k such that Solve inverse variation problems Unlike direct variation, where k > 0 and k increases as y increases. Inverse variation is the opposite. As one variable increases, the other variable decreases. Slide 7.8-10

  11. Suppose y varies inversely as the square of x. If y = 5 when x = 2, find y when x = 10. EXAMPLE 3 Using Inverse Variation Solution: Slide 7.8-11

  12. If the cost of producing pairs of rubber gloves varies inversely as the number of pairs produced, and 5000 pairs can be produced for $0.50 per pair, how much will it cost per pair to produce 10,000 pairs? EXAMPLE 4 Using Inverse Variation Solution: Slide 7.8-12

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